We present a novel approach to adaptive and progressive image transmission, based on the decomposition of an image into compositions and superpositions of monovariate functions. The monovariate functions are iteratively constructed and transmitted, one after the other, to progressively reconstruct the original image: the progressive transmission is performed directly in the 1D space of the monovariate functions and independently of any statistical properties of the image. Each monovariate function contains only a fraction of the pixels of the image. Each new transmitted monovariate function adds data to the previously transmitted monovariate functions. After each transmission step, by using the updated monovariate functions the image is reconstructed with an increased resolution. Finally, once all the monovariate functions have been transmitted, the original image is reconstructed exactly. This approach is characterized by its flexibility and robustness to packet loss: any numbers of intermediate transmissions and reconstructions are possible, and in case of packet loss, the global appearance of the transmitted image is preserved. Moreover, the intermediate images can be reconstructed at any resolution, and for any number of intermediate reconstructions, the original image will be exactly reconstructed. Finally, the quantity of data to be transmitted only depends on the image size and is independent of the number of intermediate reconstructions. Our main contributions are the modification of the decomposition scheme defined by the Kolmogorov superposition theorem to enable multiresolution image reconstructions and its application for progressive image transmission, using successively increasing resolutions. We illustrate this approach on several images and evaluate the reconstruction quality, decomposition flexibility, and error resilience during transmission.